Degeneration of 7-Dimensional Minimal Hypersurfaces Which are Stable or Have a Bounded Index

Abstract

A 7-dimensional area-minimizing embedded hypersurface \(M^7\) will in general have a discrete singular set, and the same is true if M is locally stable provided \({\mathcal {H}}^6(\textrm{sing}M) = 0\). We show that if \(M_i^7\) is a sequence of 7D minimal hypersurfaces which are minimizing, stable, or have bounded index, then \(M_i \rightarrow M\) can limit to a singular \(M^7\) with only very controlled geometry, topology, and singular set. We show that one can always “parameterize” a subsequence \(i'\) with controlled bi-Lipschitz maps \(\phi _{i'}\) taking \(\phi _{i'}(M_{1'}) = M_{i'}\). As a consequence, we prove the space of smooth, closed, embedded minimal hypersurfaces M in a closed Riemannian 8-manifold \((N^8, g)\) with a priori bounds \({\mathcal {H}}^7(M) \leqq \Lambda \) and \(\textrm{index}(M) \leqq I\) divides into finitely-many diffeomorphism types, and this finiteness continues to hold if one allows the metric g to vary, or M to be singular.